Dynamic Pricing Control for Open Queueing Networks

Pricing control is an important problem in service systems and it aims to control customer behaviors through an economic way, instead of administrative commands. In this paper, we study a dynamic pricing and service rate control problem in an open Jackson network with limited capacity. The goal is to determine the optimal admission prices and the optimal service rates at every state such that the long-run average social welfare is maximized. The original problem is decomposed into a rate-setting problem plus a price-setting problem. To solve the rate-setting problem, we derive a difference formula based on the sensitivity-based optimization theory. When the cost rate function is convex in service rates and the value rate function is concave in arrival rates, we decompose the rate-setting problem into a series of convex optimization subproblems. When the rate functions have linear structure, these subproblems are even simpler and a bang-bang control is optimal. For the price-setting problem, we determine the state-dependent prices so as to induce the optimal arrival rates obtained by the rate-setting problem. We propose a recursive algorithm to numerically compute the conditional expected delays at every state. Finally, we conduct numerical experiments to explore the optimality properties and some useful insights for this dynamic pricing control problem.